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A component map tuning method for performance prediction and diagnostics of gas turbine compressors
TSOUTSANIS, Elias <http://orcid.org/0000-0001-8476-4726>, MESKIN, Nader, BENAMMAR, Mohieddine and KHORASANI, Khashayar
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TSOUTSANIS, Elias, MESKIN, Nader, BENAMMAR, Mohieddine and KHORASANI, Khashayar (2014). A component map tuning method for performance prediction and diagnostics of gas turbine compressors. Applied Energy, 135, 572-585.
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A component map tuning method for performance prediction anddiagnostics of gas turbine compressors
Elias Tsoutsanisa, Nader Meskina,∗, Mohieddine Benammara, Khashayar Khorasanib
aDepartment of Electrical Engineering, College of Engineering, Qatar University, Doha, QatarbDepartment of Electrical and Computer Engineering, Concordia University, Montreal, Canada
Abstract
In this paper, a novel compressor map tuning method is developed with the primary objective of improving
the accuracy and fidelity of gas turbine engine models for performance prediction and diagnostics. A new
compressor map fitting and modelling method is introduced to simultaneously determine the best elliptical
curves to a set of compressor map data. The coefficients that determine the shape of the compressor map
curves are analyzed and tuned through a multi-objective optimization scheme in order to simultaneously
match multiple sets of engine performance measurements. The component map tuning method, that is
developed in the object oriented Matlab Simulink environment, is implemented in a dynamic gas turbine
engine model and tested in off-design steady state and transient as well as degraded operating conditions. The
results provided demonstrate and illustrate the capabilities of our proposed method in refining existing engine
performance models to different modes of the gas turbine operation. In addition, the excellent agreement
between the injected and the predicted degradation of the engine model demonstrates the potential of
the proposed methodology for gas turbine diagnostics. The proposed method can be integrated with the
performance-based tools for improved condition monitoring and diagnostics of gas turbine power plants.
Keywords: Component map, Model adaptation, Performance prediction, Gas turbine, Condition
monitoring
Highlights
• A method for fitting rotated elliptical curves to compressor performance map data is presented.
• The proposed fitting method is integrated into a dynamic model of a gas turbine.
• The performance of the method is tested in steady state and transient conditions of gas turbine.
• The proposed method is used to diagnose compressor fouling from transient data.
∗Corresponding authorEmail address: [email protected] (Nader Meskin)
Preprint submitted to Applied Energy August 2, 2014
• The maintenance cost attributed by the accuracy of the proposed method is assessed as compared to
other methods.
Nomenclature
Symbols
a semi-major axis of an ellipse
A matrix of elements α
b semi-minor axis of an ellipse
f flow rate
g generic form of map’s sub/coefficients
m corrected mass flow rate
m mass flow rate
m′ mass flow rate from plenum
n total number of operating points per speed line
N corrected shaft rotational speed
p pressure
p′ pressure from plenum
P power
q total number of speed lines
sm surge margin
T temperature
T ′ temperature from plenum
u ambient and operating conditions vector
x coordinate of an ellipse in x-axis
X component characteristics vector
y coordinate of an ellipse in y-axis
Y measurement vector
Greek
α element of matrix A
Γ mass flow capacity
ε average prediction error
η isentropic efficiency
θ angle of rotated ellipse
2
π pressure ratio
σ spread, smooth parameter
Subscript
act actual
amb ambient
c compressor
cl clean
d diffuser
deg degraded
des design point
f fuel
i order of the polynomial, number of the speed lines
inj injected
j number of the operating points
max maximum
pred predicted
pt power turbine
r reference engine
s steady state
surge surge point
t turbine
tr transient
w weighted
0 fixed coordinate, θ=0
2 compressor inlet
3 compressor exit
4 combustor exit
5 turbine exit
6 power turbine exit
7 exhaust
1. Introduction1
Gas turbine performance simulation, diagnosis and prognosis are strongly dependent on detailed under-2
standing of the engine component behavior. Typically the bevahior of an engine component is represented3
3
by performance maps which are the Original Equipment Manufacturer’s (OEM) proprietary design infor-4
mation. At the off-design conditions, the quality of engine component maps is vital for the accuracy of gas5
turbine performance and diagnostic models. Special interest is given to compressors since they can generate6
various operational problems such as surge, stall and flutter, although their operating line is determined7
by the turbine characteristics. Maps can also be determined by flow analysis schemes such as the Stream8
Line Curvature (SLC) method [1] or high fidelity Computational Fluid Dynamics approaches (CFD) if the9
geometry of the compressor is known.10
Gas turbine users do not have access to the context of such maps and their involvement is only limited11
to the use of tailored to customer decks for performance computations. The above limitations have recently12
motivated gas turbine research community to explore alternative methods [2],[3],[4] for representing the13
compressor maps in order to improve the accuracy of the performance prediction.14
Some researchers have focused on the map representation itself, while others have implemented compres-15
sor map methods further in performance models of gas turbines. Kurzke [5] introduced auxiliary coordinates16
(beta lines), having no physical significance, which are superimposed in order to address the non-uniqueness17
and ill-conditioning issues of the compressor map shapes. Jones et al. [6] and Sethi et al. [7] introduced18
quasi-physics-based backbone compressor mapping approaches. The work by Drummond and Davison [8]19
examined the shape variance of the compressor maps for a wide range of compressor shapes and related20
them to the physical processes. A promising data-based method for the component performance which uses21
Back Propagation Neural Networks (BPNN) has been developed by Yu et al. [2]. The general regression22
neural networks (GRNN) and Multi Layer Perception (MLP) approaches that have been suggested by M.23
Gholamrezaei and K. Ghorbanian [9] provide good prediction of the compressor map shape. The accuracy24
and performance of the above methods depend on the quality and quantity of the available engine data from25
the engine manufacturers or the engine users.26
One of the most commonly used methods involves scaling and shifting the shape of a similar compressor27
map through optimization techniques such that it matches the targeted engine measurements. Such a28
scaling method was proposed by Kong et al. [10] on the assumption that the implemented map has a very29
similar shape to the actual map. A combined hybrid approach that was developed by Kong et al. [3] takes30
advantage of Genetic Algorithms (GA) for determining the coefficients of the polynomials used in the system31
identification method. Recently Li et al. [4] have suggested a unique set of scaling coefficients for each line32
of the constant speed and efficiency to capture nonlinear effects. This method yields a higher accuracy than33
the traditional scaling methods at operating points away from the design point of the engine.34
Although the advantages and benefits of the above approaches in engine performance prediction are35
extensive, the trade-offs between key parameters such as the operating range, accuracy, complexity and36
computational time are still debated and worth further investigation. Data-based approaches such as Neural37
Networks (NN) [2, 9] and GA [10, 11] might provide improved prediction accuracy with the compressor map38
4
data and/or engine measurements; however they suffer from the extensive training and non-uniqueness of39
the solution, respectively. On the other hand, in diagnostics systems and approaches both GA and NN can40
be designed to be robust due to performing prior signal processing operations [12]. Since transient data41
are normally acquired online at reasonably high rates the former methods cannot be used effectively when42
integrated into a dynamical engine model. This is specially the case for real-time performance prediction of43
a gas turbine. Therefore, it is essential to develop a robust, formally validated, generic and computationally44
efficient approach for representing a compressor map that when empowered with model adaptation will be45
able to predict a gas turbine’s performance at off-design steady state and transient conditions.46
In this paper, a compressor map generation method for improving the accuracy of the gas turbine47
performance prediction and diagnostics is developed. In contrast to the earlier work [13], where the shape of48
a compressor map was expressed by mathematical equations of an ellipse with fixed center and no rotation,49
this study proposes a more sophisticated and accurate approach by considering rotation of the ellipses50
and transformation of its coordinates. This method is integrated into a dynamic engine model that is51
developed in the Matlab/Simulink environment. The dynamic engine model itself and its validation against52
the gas turbine simulation software PROOSIS [14] were the subject of our earlier work [15]. Moreover, the53
application is not only tested to the steady state off-design operation [16], but is also extended to transient54
operation for healthy and degraded conditions. Finally, a simplified techno economic assessment based on55
the work of Aretakis et al. [17] is carried out for evaluating the effects that the accuracy of our method have56
on the cost of gas turbine maintenance when the compressor is degrading.57
2. Methodology58
Axial compressor performance maps are used in the gas turbine thermodynamic models for estimation59
of key component parameters such as the pressure ratio πc, the corrected mass flow rate mc =(mc×
√T 2
p2
)60
and the isentropic efficiency ηc at several corrected rotational speeds N . A typical compressor map available61
from the literature [18] has been digitized and reproduced as shown in Fig. 1. Generating compressor maps62
for low speed regions, that is below 50% of the corrected rotational speed, is another research topic that has63
been addressed by other researchers [6], [19] and is beyond the scope of this study.64
The objective of map generation approaches is to determine mathematical expressions that could accu-65
rately capture the shape of the map. This is performed by relating the corrected mass flow rate mc and the66
isentropic efficiency ηc with the pressure ratio πc and the corrected rotational speed N , i.e. mc = f(πc, N)67
and ηc = g(mc, N). Another way of expressing the efficiency ηc = h(πc, N) is by expressing it as a function68
of the pressure ratio πc, which is beneficial for high rotational speed lines that are almost vertical. A detailed69
description of our proposed method follows in the next subsections.70
5
0 10 20 30 40 50 60 700
5
10
15
20
0.8650.86
0.87
0.84
0.80.75
0.7
0.5260.630.737
0.790.842
0.8610.896
0.948
1
1.053
Mass Flow Rate (kg/s)
Pressure
Ratio
LM2500 Compressor Performance Map
Figure 1: Compressor performance map as reproduced from [18].
2.1. Map Fitting71
The process of map fitting commences with a reference map that is available either from the open72
literature or constructed from the operational data. Having the reference compressor map, the objective73
is to fit the available data with a single mathematical expression, which should be of the same form for74
every speed line. The accuracy of the fitting procedure depends on several factors, such as the complexity75
of the mathematical model chosen, the quality of the data, the threshold of the tolerance criterion and the76
objective function that is used in the optimization, if any. Several maps from the literature have been used77
to test the validity of the proposed method but due to space limitations in this work we will only present78
the accuracy of the method as it is applied to the compressor map shown in Fig. 1. After an extensive79
review of several methods (polynomials, neural networks [2],[9], etc.) for representing the pressure ratio πc80
and the efficiency ηc as a function of mc and N , the most mathematically robust approach is determined81
where each line belongs to an elliptic curve. The equation, adjusted for the πc versus mc map, is given by82
(mc0 − x0
aπc
)2
+
(πc0 − y0
bπc
)2
= 1 (1)
where aπc and bπc denote the semi-major and the semi-minor axes of the ellipse, respectively. In addition83
mc0 and πc0 denote the corrected mass flow rate and the pressure ratio when the ellipse is fixed at (x0, y0),84
which represents the center coordinates of the ellipse, respectively. Taking into consideration that each85
ellipse is free to rotate, at an angular value of θπc , the new coordinates of the ellipse (mc, πc) are now given86
by87
mc = mc0 cos(θπc)− πc0 sin(θπc) (2)
6
πc = mc0 sin(θπc) + πc0 cos(θπc) (3)
Similarly for the efficiency the governing equation is given by88
(mc0 − x0
aηc
)2
+
(ηc0 − y0
bηc
)2
= 1 (4)
where aηc and bηc denote the semi-minor and the semi-major axes of the ellipse, respectively, and mc0 and89
ηc0 denote the corrected mass flow rate and the efficiency. Once again, rotating the ellipse at an angle of90
θηc yields the compressor’s isentropic efficiency ηc as91
ηc = mc0 sin(θηc) + ηc0 cos(θηc) (5)
where eq. (2) is used to determine mc since the range of the corrected mass flow is identical for both πc vs.92
mc and ηc vs. mc maps.93
The start and the end of the map generated for each speed line are limited according to the variation of94
each coefficient with respect to the corrected rotational speed N . The distribution of the points at which95
the surge occurs πcsurge is expressed as a 2nd order polynomial function of the corrected rotational speed N ,96
as follows:97
πcsurgei = aNi2 + bNi + c (6)
where a, b, c are the coefficients of the equation and i denotes the corresponding speed line of the surge point98
πcsurge . Assuming a constant surge margin sm of 20%, the maximum pressure ratio πcmax of each speed line99
is then given by:100
πcmaxi =
(πcsurgei
(1 + sm)
)(7)
This surge limiter prevents the speed lines from exceeding their corresponding maximum pressure ratio.101
Three approaches of varying complexities have been proposed for fitting the πc versus mc map data with102
the following assumptions on the ellipses, namely103
Approach 1. Center at (0, 0) and no axes rotation,104
Approach 2. Center at (0, 0) and rotation of the axes at an angle of θ,105
Approach 3. Center at (x0, y0) and rotation of the axes at an angle of θ.106
Similar approaches are considered for the ηc versus mc map apart from the first one, where x0 is assumed107
to be the mid point of each curve as follows:108
Approach 1. Center at (x0, 0) and no axes rotation.109
7
bπc
aπc(0, 0)
πc0
mc0
Speed Line
(a) πc vs. mc - Approach 1.
Speed Line
bπc
aπc
(0, 0)
πc0
mc0θπc
mc
πc
(b) πc vs. mc - Approach 2.
)
Speed Linebπc
aπc
(0, 0)
πc0
mc0
θπc
mc
πc
(x0, y0)
(c) πc vs. mc - Approach 3.
Efficiency Linebηc
aηc(0, 0)
ηc0
mc0(x0, 0)
(d) ηc vs. mc - Approach 1.
Efficiency Line
bηc
aηc
(0, 0)
ηc0
mc0
θηc
mcηc
(e) ηc vs. mc - Approach 2.
Efficiency Line
bηc
aηc
(0, 0)
ηc0
mc0
θηc
mcηc
(x0, y0)
(f) ηc vs. mc - Approach 3.
Figure 2: Elliptical curve parameters for various fitting approaches.
Approach 2. Center at (0, 0) and rotation of the axes at an angle of θ,110
Approach 3. Center at (x0, y0) and rotation of the axes at an angle of θ.111
A graphical illustration of the suggested elliptical curve fitting approaches along with the important pa-112
rameters of eqs. (1)-(5) is shown in Fig. 2. Furthermore, a family of the compressor map fitting approaches as113
suggested by Gholamrezaei and K. Ghorbanian [9], and Yu et al. [2] are implemented according to the same114
objectives. The fitting methods employed here are the GRNN approach with constant and variable spreads115
σ [9] and a typical BPNN method [2]. The GRNN approach [9] and the BPNN method [2] have been tested116
by implementing the Matlab’s GRNN functions and the NN toolbox [21], respectively. The value of each117
coefficient for the Approaches 1, 2 and 3 has been determined by integrating the above elliptical functions118
in one of the Matlab’s minimization algorithms (specifically the [fminsearch] [21]). The initial conditions for119
these coefficients did not require any expert knowledge and could be easily set based on the elliptical curve120
properties as shown in Fig. 2. The application of the proposed and available methods for the selected map121
is shown in Fig. 3.122
It is observed from Figs. 3a and 3b that the Approach 1 yields similar results as in the GRNN method123
with constant spread σ. When approaching the high corrected rotational speed N region, the accuracy124
of both methods decreases and they are not accurately capturing the curvature of each speed line. In the125
second group of fitting approaches, as shown in Figs. 3c and 3d, Approach 2 is more accurate than Approach126
1 and the BPNN fitting method. Once again, at the high rotational speed N region the BPNN performance127
is not as accurate; nevertheless it is more accurate than Approach 1 and GRNN with fixed spread.128
8
20 40 60
5
10
15
20
Corrected Mass Flow Rate mc
Pre
ssu
reR
ati
oπ
c
πc vs. mc
Compressor Data GRNN fixed σ Approach 1
(a) πc-Fitting (Approach 1 and GRNN fixed σ).
20 40 60
0.7
0.8
Corrected Mass Flow Rate mc
Isen
trop
icE
ffici
ency
ηc
ηc vs. mc
(b) ηc-Fitting (Approach 1 and GRNN fixed σ).
20 40 60
5
10
15
20
Corrected Mass Flow Rate mc
Pre
ssu
reR
atioπ
c
πc vs. mc
Compressor Data BPNN Approach 2
(c) πc-Fitting (Approach 2 and BPNN).
20 40 60
0.7
0.8
Corrected Mass Flow Rate mc
Isen
trop
icE
ffici
ency
ηc
ηc vs. mc
(d) ηc-Fitting (Approach 2 and BPNN).
20 40 60
5
10
15
20
Corrected Mass Flow Rate mc
Pre
ssu
reR
atioπ
c
πc vs. mc
Compressor Data GRNN variable σ Approach 3
(e) πc-Fitting (Approach 3 and GRNN variable σ).
20 40 60
0.7
0.8
Corrected Mass Flow Rate mc
Isen
trop
icE
ffici
ency
ηc
ηc vs. mc
(f) ηc-Fitting (Approach 3 and GRNN variable σ).
Figure 3: Compressor performance map fitting methods.
9
0.6 0.7 0.8 0.9 10
0.5
1
Corrected Rotational Speed NNor
mV
alu
eof
Coeffi
cien
ts
aπc bπcθπc
Figure 4: Variation of the proposed map fitting coefficients as a function of the corrected speed N for the πc versus mc map.
It is evident from the final group of fitting approaches that are shown in Figs. 3e and 3f, that Approach129
3 provides a very good agreement with the compressor map data; as does the GRNN method with variable130
spread. We are now in a position to evaluate the above proposed approaches in terms of their computational131
cost and fitting performance given that they have different complexities. The common parameters of each132
ellipse in all the proposed approaches are a and b. In addition, the angles θηc and θπc of the rotated ellipses133
are employed in both the Approaches 2 and 3. The third approach has also the center coordinate parameters134
(x0, y0) and Approach 1 has an additional parameter x0 for the efficiency map. The only important parameter135
to be computed in both GRNN methods is the spread or the smoothing parameter σ.136
All the above parameters or coefficients should be expressed as functions of the corrected rotational137
speed N . This process results in a number of sub-coefficients depending on the type of the function. The138
propagation of the coefficients with respect to N should be captured by smooth and simple functions that139
will not give rise to ill-conditioned extrapolation affecting the accuracy of the targeted optimization solution.140
Each coefficient of the suggested elliptical approach is now expressed as a polynomial function of the corrected141
rotational speed in the generic form as142
g(N) = g1Ni + · · ·+ giN + gi+1 (8)
where g denotes as one of map coefficients and i denotes the order of the polynomial function. For instance,143
in the second approach, the coefficient aπc is now expressed as144
aπc(N) = aπc1N3 + aπc2N
2 + aπc3N + aπc4 (9)
which is a 3rd order polynomial with a total number of 4 sub-coefficients. In Approach 2 where the coefficients145
for the pressure map (aπc , bπc , θπc) are expressed as a function of the speed, as shown in Fig. 4, this results146
in 13 sub-coefficients for the πc versus mc map.147
The above process is performed for all the three proposed fitting approaches using the Matlab’s curve148
fitting toolbox. The total number of the sub-coefficients for each proposed approach along with the fitting149
10
errors of the available methods are tabulated in Table 1. For the GRNN method with a constant spread150
this parameter can be expressed as a linear function of the corrected rotational speed N with only two151
sub-coefficients, i.e. σ = σ1N +σ2. The qualitative evaluation and comparison of the above compressor map
Table 1: Evaluation of various compressor map fitting methods.
Method CoefficientCoefficient
as f(N)
Sub-
coefficients
No of
Sub-coef.
Fit Error
mc, ηc (%)
App.1 [13]aπc , bπc ,
aηc , bηc , x0ηc
Polynomialaπc1...4 , bπc1...4 ,
aηc1...4 , bηc1...4 , x0ηc1...4
20 6.5, 0.89
App. 2aπc , bπc , θπc ,
aηc , bηc , θηc
Polynomialaπc1...4 , bπc1...5 , θπc1...4 ,
aηc1...3 , bηc1...4 , θηc1...3
23 2.9, 0.54
App. 3
aπc , bπc , θπc ,
x0πc, y0πc
, aηc , bηc ,
θηc , x0ηc, y0ηc
Polynomial
& Splines
aπc1...8 , bπc1...9 , θπc1...10 ,
x0πc1...13, y0πc1...9
, aηc1...10 ,
bηc1...8 , θηc1...12 , x0ηc1...14,
y0ηc1...7
100 2.2, 0.41
BPNN [2] - - - - 3.8, 1.41
GRNN
constant σ [9]σ Linear σ1...2 2 6.5, 0.71
GRNN
variable σ [9]σ - - - 0.5, 0.24
152
fitting methods highlight several trade-offs among their performance, robustness and further implementation153
in gas turbine models. Generally, the GRNN with fixed spread parameter is a good candidate along with154
the Approach 1, since they are able to extrapolate data in a compressor map, but both suffer in terms155
of accuracy. Approach 3 yields the most accurate results among the proposed elliptical fitting approaches156
due to the fact that there is freedom in changing the center coordinates of the ellipses and simultaneously157
rotating them. The distribution of the Approach 3 coefficients, as a function of N , is highly nonlinear and158
only high order polynomials and spline curves are capable of capturing effectively this nonlinear distribution.159
This results in the use of 100 sub-coefficients for the Approach 3 as opposed to 23 that are required for the160
Approach 2 although their differences are not significant in both the pressure ratio and the efficiency maps.161
Approach 2 has a fitting error of 2.9% for mc which may be considered high to allow for an accurate162
11
prediction given that for diagnostic purposes one examines faults greater than 1%. However, this fitting163
error refers to the accumulated deviation from all the points of the reference map. This implies that the164
average fitting error in Approach 2 for mc is 0.044% corresponding to each one of the 50 data points. The165
GRNN with variable spread provides excellent results, however it has the same disadvantage as Approach166
3 in terms of its implementation in a gas turbine model, since the distribution of the spread parameter as167
a function of the speed is highly nonlinear to allow for reliable extrapolation. The excessive parameters168
corresponding to the GRNN with the variable spread has motivated the development of the rotated GRNN169
(RGRNN) and MLP approaches as presented in [9]. While accuracy is improved by the latter methods, the170
RGRNN is limited to represent compressor curves for which data are available. On the other hand, further171
implementation of the MLP approach in a dynamic adaptive engine model for performance prediction and172
diagnostics remains to be investigated.173
The above comparison is made in order to emphasize that although certain methods will always, inde-174
pendent of the map shape, outperform others in fitting the compressor map data, this goal is not the only175
measure for a successful gas turbine performance prediction and diagnostic scheme. The well-conditioned176
interpolation, extrapolation and the ability of the map fitting algorithms to account for the degradation177
factors are far more important and crucial than their absolute accuracy that is expressed in terms of the178
fitting error criterion.179
Therefore, we have concluded that Approach 2 is selected to be implemented for the engine performance180
prediction by its integration in a gas turbine model. This decision is justified by the fact that Approach181
2 yields a good balance between the accuracy that is obtained and the complexity of the mathematical182
expressions capturing the variations of each coefficient with respect to the speed. The analytical methodology183
enables one to formally control the compressor map shape in a nonlinear manner, so that the corresponding184
map generation method can replace simple lookup tables and/or externally linear-scaled maps in an engine185
model. Another advantage of our proposed fitting method, due to the analytical nature of the expressions186
employed for representing an ellipse, is the fact that initial values of the coefficients (a, b, θ) can be selected187
without any empirical knowledge of the desired maps.188
2.2. Gas Turbine Engine Model189
The proposed compressor map fitting method is now integrated into a dynamic model of a two shaft190
industrial gas turbine that is developed in Matlab/Simulink environment and validated with PROOSIS [14].191
The engine model that is developed consists of a compressor, a combustor, a compressor turbine and a power192
turbine as shown in Fig. 5. The components are represented by a set of suitable component maps from193
PROOSIS, although the compressor map that is used is the one shown Fig. 1.194
The engine simulations performed consist of two modes, the steady state and the transient. Steady195
state simulations are performed for a scheduled variation of the rotational speed N and through an iterative196
12
Compressor
Combustor
Turbine Power Turbine
Exhaust
Tamb
pamb
ηd
Ambient
p′3
T2
p2
ηd
N
m3
T3
Pc
Thermo-dynamics
m′3
m3
T3
p′3
T ′3
VolumeDynamics
ff
ff
T ′3
p′3
m′4
m′3
T4
p4
p′5
T4
p4
N
m5
T5
Pt
Thermo-dynamics
m′5
m5
T5
p′5
T ′5
VolumeDynamics
p′6
T ′5
p′5
m6
T6
Ppt
Thermo-dynamics
m′6
m6
T6
p′6
T ′6
VolumeDynamics
p′6
T ′6
m′7
Pc
Pt
N
Rotor Dynamics
Figure 5: Engine model layout in Simulink (for definition of the variables refer to the Nomenclature section).
method, that selects key component parameters (i.e. mass flow rate m, Turbine Entry Temperature (TET),197
etc.) for satisfying the mass flow and the work compatibility laws, where the model converges to the steady198
state condition. From the converged steady state condition, the values of temperatures at the compressor,199
combustor, compressor turbine and the power turbine exits, namely, T3, T4, T5, T6, are passed onto the200
transient model’s plenum volumes as initial conditions.201
The transient simulation, based on the Inter Component Volume (ICV) method [20], commences from202
the last known converged steady state condition of the model with the variation of the control vector which203
is the fuel flow ff . As can be observed from the engine layout shown in Fig. 5 plenums are introduced204
between various components to account for all the flow imbalances to occur in these volumes. These mass205
imbalances are used to evaluate the rate of the pressure increase to the engine and hence calculate the206
values of the pressures at the compressor, turbine and the power turbine plenum exits, namely, p′3, p′5, p′6.207
The difference between the power required by the compressor Pc and the power extracted from the turbine208
Pt at any given time instant yields an estimate of the rotor acceleration, and hence the rotor speed N . The209
process is repeated for the next time interval until the end of the simulation time is reached. The developed210
engine model in the Simulink, an object oriented environment, can be easily adapted to any kind of gas211
turbine configuration. A detailed description of the model used for this application can be found in our212
earlier work in [15].213
13
2.3. Model Adaptation214
Model performance adaptation is concerned with the inverse performance analysis where engine compo-215
nent characteristics are tuned until they reproduce the measured engine behavior at the same environmental216
conditions and throttle setting. Generally the engine behavior, assuming no measurement noise or bias, is217
expressed as follows:218
Y = f(X,u) (10)
where Y denotes the engine performance vector consisting of the measurements such as the pressure and219
the temperature at different engine gas path locations, namely Y = [p, T ]. The component characteristic220
vector X includes non-measurable quantities such as the corrected mass flow and the efficiency, namely221
X = [mc, ηc] and u denotes the ambient and the operating conditions vector consisting of ambient conditions222
and a parameter called handle of the engine model which acts as input to the model for engine performance223
simulations, namely u = [pamb, Tamb, handle]. Depending on the simulation approach selected, the handle224
can be either useful power output from the power turbine Ppt, TET, corrected rotational speed N or any225
other quantity.226
The measured engine behaviour is represented either by the field data of a service engine or through227
simulations of a different engine model. For conducting testing of our proposed method, the reference engine228
is an engine model that uses the compressor performance map of Fig. 1. In contrast to the map generation229
procedure that is employed by the engine model, the reference engine employs the map of Fig. 1 as simple230
lookup tables for determining the corrected mass flow rate and the isentropic efficiency of the compressor.231
Engine Model
Amb.& Op.
conditions
Componentparameters
Map
Engineperformanceparameters
Reference engine
OFMin?
Optimization
Stop
Yes
No
u
u
X
Y
Yr
Tune Component Map
Figure 6: Flow chart of adaptive performance simulation.
The adaptive simulation algorithm that is depicted in Fig. 6 attempts to tune the component charac-232
teristics vector X so that the difference between the performance vector of the engine model Y and the233
14
reference engine Yr is minimized. For this study, the difference between the predicted Y and the observed234
Yr measurements can be evaluated by means of an Objective Function (OF ) that is defined as follows:235
OF =‖A‖F =
√√√√ q∑i=1
n∑j=1
|αi,j |2 (11)
=
√√√√ q∑i=1
n∑j=1
∣∣∣∣Yi,j −Yri,j
Yri,j
∣∣∣∣2 (12)
where n denotes the total number of operating points corresponding to q number of different corrected236
rotational speed lines. The Frobenius norm ‖A‖F of an m × n matrix A is defined as the sum of the237
absolute squares of its elements αi,j. The element αi,j denotes the difference between the predicted Yi,j and238
the measurable performance vector Yri,j , divided by Yi,j . Since the above adaptive performance simulation239
covers multiple operating points, the objective function was modified to accommodate for such a feature.240
The number of the measured parameters to be matched depends on the test case itself, however for this241
work all the measured parameters are used in a single optimization problem.242
The criterion OF is minimized by implementing one of the Matlab’s built-in nonlinear unconstrained243
optimization algorithms [fminsearch] [21], which is an implementation of the Nelder Mead algorithm [22].244
This algorithm is most effective in exploring the neighborhood of the starting point and converging to a245
local minimum of OF . However, mathematically multiple solutions might be possible since the algorithm246
employed here might not always provide the global minimum of OF . This limitation is addressed by initially247
matching the operating points that are close to the design point of the engine and then moving further away248
from the design point. Another measure taken to address the former limitation is to set the optimization249
without any constraints on the sub-coefficients values in order to increase the search space. The last criterion250
for successful optimization is that convergence of the solution should occur before reaching the maximum251
number of iterations. The above steps ensure that the solution obtained is the global minimum of the OF .252
It should be emphasized that the proposed method assumes the existence of an initial map shape. There253
is no similarity or closeness requirement between the initial map selected for the engine model and the254
unknown compressor map of the reference test engine. This is supported by the fact that the method is255
capable of regenerating any shape of the compressor map since the map curves are analytically controlled.256
The number of the sub-coefficients utilized has to do only with the fidelity by which a compressor map257
shape is generated and is not related to the number of gas path measurements that are to be matched. The258
adaptation procedure is now described as follows:259
1. Select an initial compressor map from the open literature or any other available source.260
2. Scale the map linearly so that the compressor map parameters at 100% of their nominal value261
(mcmap , ηcmap , πcmap) satisfy the design point performance of the reference engine (mcdes , ηcdes, πcdes).262
15
3. Fit the elliptical curves to the linear scaled initial map in order to determine the coefficients aπc , bπc ,263
θπc , aηc , bηc , θηc .264
4. Express the coefficients with respect to the corrected rotational speed N and determine the values of265
the 23 sub-coefficients. Note that one may choose to match a map shape that is uniformly smooth and266
not so mathematically challenging as the one shown in Fig. 3 in order to utilize high order polynomials267
for expressing several map coefficients. This is something that is facilitated by the optimizer since the268
algorithm has the freedom to assign zero values to the sub-coefficients, and therefore reduce the order269
of the polynomials.270
5. Integrate the map fitting equations along with the sub-coefficients of the initial scaled map in a gas271
turbine engine model.272
6. Utilize the gas path measurements of a reference engine and let the adaptation process tune the sub-273
coefficients to generate an adapted map for matching all the measurements. Note that before the274
adaptation process is employed the simulated engine measurements will have significant deviations275
from those of the reference engine, and in some cases convergence of the engine model might not be276
possible before invoking the optimization process.277
7. The final adapted map is a compressor map that is capable of generating the same results as the gas278
path measurements available. The generated map is only an approximation to the unknown map since279
there are many uncertainties such as measurement noise, humidity, etc., that are not accounted for.280
Specifically, for the case studies examined in this paper the initial values of the sub-coefficients are281
determined by the fitting procedure performed earlier to a typical reference compressor map. These are282
passed onto the compressor map generation procedure as shown in Fig. 7, along with the measured corrected283
rotational speed Nr, in order to determine the coefficients of the map (aπc , bπc , θπc , aηc , bηc , θηc). Once the284
coefficients are calculated, the pressure ratio πcr , which is a known measurable parameter of the reference285
engine, is used for determining the corrected mass flow mc by solving eqs. (1), (2) and (3). The corrected286
mass flow mc is then used in the same manner as in πcr to calculate the compressor’s isentropic efficiency287
ηc from eqs. (4), (5).288
Both mc and ηc form the component characteristics vector X which is then used in the remaining ther-289
modynamic computations of the engine model. The key optimization parameters, such as the number of290
iterations and the tolerance criterion are specified accordingly. Once the optimization process converges, the291
new set of sub-coefficients are passed onto the engine model. The developed algorithm can be executed both292
for external and internal adaptive simulations. In case of internal adaptive simulation, the optimization of293
sub-coefficients takes place simultaneously with the iterative computation of the compatibility thermody-294
namic equations of the engine model.295
16
aπc1
aπc4
aπc
Map
eq.(1)eq.(2)eq.(3)
Nr πcr
mcbπc1
bπc5
bπc
θπc1
θπc4
θπc
aηc1aηc3
aηc
eq.(4)eq.(5)
ηc
X
bηc1bηc4
bηc
θηc1θηc3
θηc
Figure 7: Flow chart of compressor map generation.
The data driven nature of the proposed method implies that the accuracy of the map generated depends296
on the number of operating points that are available from an engine with an unknown map. However, it297
should be noted that the generated map is an accurate representation of the unknown map mainly for the298
region for which the data points are available. This implies that a map can be generated even when a small299
number of operating points or a set of points distributed in the same speed line are available. The remaining300
area of the generated map can be considered as a reasonable estimate of the unknown map according to the301
extrapolation capabilities of the method. The ideal case will be when the targeted operating points cover a302
wide area of the map. In such a case the generated map is a more accurate representation of the unknown303
map.304
2.4. Adaptive Diagnostics305
An additional feature of the model adaptation is that it can be applied not only for performance306
simulation but also for gas turbine diagnostics. Below we employ the model adaptation scheme for per-307
forming engine diagnostics as shown in Fig. 8. The component vector X consists of mass flow capacity308
Γc = mc/mcdes and isentropic efficiency ηc. Compressor degradation of the reference engine is repre-309
sented by the deviation of these parameters from their healthy values, i.e. ∆Γcinj = 100(Γcdeg − Γccl)/Γccl ,310
∆ηcinj = 100(ηcdeg − ηccl)/ηccl . The above deviations injected to the reference engine form the deviation311
vector ∆Xinj .312
Consequently, the reference engine will operate at degraded conditions and will produce a new set of313
degraded measurable parameters Yrdeg . The task of the adaptation scheme in this case is to determine the314
17
u
∆Xinj
Degraded Reference Engine
Degraded Engine Model
Map TuningXdeg
No
YesOF=0 Stop
Yrdeg
Ydeg
Figure 8: Flow chart of engine model adaptive diagnostics.
rate of the degradation that is injected in the reference engine, by tuning the component map of the engine315
model as described earlier. Therefore, the engine model matches the targeted degraded measurements of316
the reference engine, by a new set of components parameters that form the degraded vector Xdeg. The317
earlier adaptation of the model is in fact a training phase for fault diagnosis since it acts as the reference318
point for the engine healthy condition. Therefore, the difference between the degraded and the healthy set319
of component vectors (Xdeg,Xcl) determines the severity of the degradation320
∆Xpredi,j = 100
(Xdegi,j −Xcli,j
Xcli,j
)(13)
In order to assess how effective the prediction results are a diagnostic index (DI) is defined as follows321
DI = 100
(1
1 + ε
)(14)
where ε is the average error in terms of the characteristic vector X which is the output of the compressor322
map. In contrast to the gas path analysis (GPA) index used in the GPA methods [23], the one utilized323
here assesses the effectiveness of the prediction based on the output of the compressor map which forms the324
characteristic vector X. Consequently, the accuracy by which a map is tuned to meet the degraded gas path325
measurements is evaluated according to326
ε =
q∑i=1
n∑j=1
(∆Xpredi,j
−∆Xinji,j
∆Xinji,j
)qn
(15)
For multiple operating points, n denotes the total number of points corresponding to q number of different327
corrected rotational speed lines. The simulation case studies that have been carried out are now described328
in the following section.329
3. Case Study Description330
Our proposed adaptation approach is implemented in a dynamic engine model that is developed in331
the Matlab/Simulink environment and is evaluated and analyzed for the steady state and the transient332
18
conditions. As described earlier, the reference engine is a similar model with simple lookup tables whereas333
the engine model utilizes our proposed map generation process. The performance specifications of the334
reference engine are provided in Table 2.335
Table 2: Performance specification of the reference engine.
Parameter Value Units
Power 3.4 MW
Pressure ratio 10.8
Thermal efficiency 38 %
Exhaust flow rate 34 kg/s
One of the prerequisites for successful model adaptation is that the engine measurable parameters are336
directly influenced by the component characteristic parameters to be adapted, otherwise non-physical devi-337
ations may appear on the component parameters which are not the cause of the initial difference between338
the predicted model and the measured values. Our primary objective of presenting the case studies is to339
evaluate the achievable accuracy improvement of our proposed method that incorporates the compressor340
map tuning and takes into consideration the above pre-requisite. Therefore, the selection of the inlet and the341
outlet measurements of the compressor are well justified. The list of the selected measurable parameters for342
conducting the adaptive performance simulations is given in Table 3. The nominal operating point that is343
chosen as the model design point for this configuration is set at 3.4 MW with the shaft corrected rotational344
speed N set as the handle of the engine.345
Table 3: Engine performance measurable parameters.
Symbol Parameter Units
p2 Compressor Inlet Pressure Pa
T2 Compressor Inlet Temperature K
p3 Compressor Discharge Pressure Pa
T3 Compressor Discharge Temperature K
Nact Shaft Rotational Speed rpm
Four case studies are conducted. The target measurements in the first case study are the deck data that346
represent the off-design steady state performance of the reference engine, when the corrected rotational speed347
N is reduced from 100% to 55% of its nominal value, and with the selected compressor map implemented348
19
as a lookup table.349
The second case study focuses on the performance of the proposed adaptation method when the deck350
data are the outcome of the transient performance of the reference engine. The effect that the initial map351
shape has on the accuracy of the proposed method is also assessed. This is accomplished by using a different352
initial compressor map shape than the one in Fig. 1, and is available from the PROOSIS gas turbine simu-353
lation software [14]. Once again, the initial map seen in Figs. 9 and 10, is used for fitting elliptical curves354
and determining the values of the sub-coefficients. Based on this map the proposed method is integrated355
into an additional engine model which is tested in transient conditions.356
5 10 15 20 25 30 35
5
10
Corrected Mass Flow Rate, mc
Pressure
Ratio,πc
Reference Data
Fit Approach 2
Figure 9: πc vs. mc map from PROOSIS [14]
5 10 15 20 25 30 35
0.7
0.8
Corrected Mass Flow Rate, mc
Isen
trop
icE
ffici
ency
,ηc
Reference Data
Fit Approach 2
Figure 10: ηc vs. mc map from PROOSIS [14]
In the third case study the goal of the engine model is to match a set of reference engine measurements357
when degradation is injected in the compressor of the latter. Therefore, the method’s prediction capability358
is not only evaluated for improving the performance simulation of the model but also for estimating the359
severity of the degradation that is injected in the reference engine.360
In addition, a typical Linear Scaling (LS) method as suggested by Kong et al. [3] and the Non Linear361
Scaling (NLS) method recently developed by Li et al. [4] have both been adopted and tested for facilitating362
their comparison with the proposed method. Both existing methods are integrated in a similar engine363
model based on the compressor map shape of Fig. 1. In terms of the optimization, both existing methods364
employed the Matlab [fminsearch] [21] algorithm and not the GA optimization scheme that the authors of365
[3], [4] have implemented in their work. This is done intentionally since uncertainty or improved accuracy366
20
that is provided by different optimization schemes should be filtered out in order to focus purely on the367
capability of each method to modify the compressor map shape, and therefore ensure that the comparisons368
between them are more realistic.369
For the transient mode cases, the fuel flow rate varies according to the fuel flow command schedule and370
the engine is decelerated and accelerated as described in Sections 4.2 and 4.3. The adaptation case study
Table 4: Simulation parameters of the adaptation case studies.
Case Mode Points IterationsFun.
Eval.
Error
Tol.
1 Steady 9 103 2× 103 10−8
2 Transient 100 104 2× 104 10−14
3 Transient 60 104 2× 104 10−14
371
parameters are provided in Table 4, where the number of maximum iterations range from 103 to 104 with372
the function evaluations being twice these values and the error tolerance ranges from 10−8 to 10−14. The373
function evaluations refer to the number of times that the optimization algorithm is allowed to evaluate the374
objective function, whereas the iterations refer to how many times this algorithm is allowed to be performed.375
The fourth case study is a techno economic assessment of each adaptation method and their associated376
cost in terms of the maintenance if adopted in a thermal power plant for performance estimation of the377
compressor degradation.378
4. Results and Discussion379
Our proposed adaptation approach is tested for both the steady state and the transient modes of the380
healthy and degraded conditions. The results for each case study are presented and discussed in the following381
subsections.382
4.1. Steady State Case Study383
4.1.1. Case 1384
385
The objective of the first case study is to evaluate the accuracy of our proposed adaptation method and386
compare it with the LS and NLS methods for a wide range of off-design steady state operations. Having the387
corrected rotational speed as the handle, the engine model is able to match the deck data of the reference388
engine at a variable level of accuracy for each method as seen in Fig. 11. It is concluded from Fig. 11 that389
21
10 15 20 25 30
2
4
6
8
10
Corrected Mass Flow Rate mc
Pressure
Ratioπc
Reference Engine Proposed AdaptationLS Method NLS method
Figure 11: Steady state operating line as predicted by various adaptation methods.
the proposed method is very accurate. The NLS method has a similar accuracy with that of the proposed390
method, but this is not the case for the LS method. The former observation is more evident from Fig. 12.391
0.5 0.6 0.7 0.8 0.9 1
0.6
0.8
1
Corrected Rotational Speed N
T3/T
3des
Reference Engine Proposed Adaptation
LS Method NLS Method
Figure 12: Normalized to design conditions compressor discharge temperature as predicted by various adaptation methods for
the steady state.
The engine model prediction error for each method is shown in Fig. 13. The engine model prediction392
error for the LS method is increasing when the operating point is far away from the design conditions. This393
is due to the fact that a major assumption for a successful adaptation in this method is that the compressor394
map to be tuned should be of very similar shape to that of the one used by the reference engine. On the395
other hand, the accuracy of the NLS method is distributed in a balanced way, and any deviations are due to396
the specific shape of the reference engine map as seen in Fig. 1 and the wide range for which that adaptation397
is pursued.398
The compressor discharge pressure p3 and the temperature T3 for the engine model show a maximum error399
in the range of -2% to 2% for the proposed method. On the other hand, the maximum engine model errors400
for the LS and the NLS methods are in the ranges of -10% to 10% and -5% to 4%, respectively. Moreover, the401
22
0.5 0.6 0.7 0.8 0.9 1
−10
0
10
Corrected Rotational Speed N
Error(%
)
T3 Proposed T3 LS T3 NLS
p3 Proposed p3 LS p3 NLS
Figure 13: Measurable parameters error of various adaptation methods for the steady state.
engine model that utilizes the proposed adaptation method has an average error in T3 which is equivalent to402
0.05K, whereas for the linear and the nonlinear scaling methods this is 15K and 2K, respectively. Figure 14403
shows the margin by which each one of the 23 sub-coefficients that control the compressor map generation404
process proposed are modified in order to match the target measurable parameters. The normalized values405
as shown in Fig. 14 simply refer to the ratio between the tuned and the initial sub-coefficients.406
5 10 15 200.8
1
1.2
1.4
No of Sub-coefficients
Nor
mal
ized
Val
ue
Initial Tuned
Figure 14: Initial and tuned map sub-coefficients for the proposed adaptation method.
Another key aspect of the entire optimization process, which involves the tuning of the sub-coefficients,407
is its rapid convergence. For this case study where 800 iterations and a total 2000 function evaluations408
were required the computation time when performed in a modern PC equipped with Intel’s i5 quad-core409
processors and 4GB of RAM memory was only 0.23 sec. The computation time for the LS and the NLS410
methods was 0.1 sec and 0.4 sec, respectively.411
4.2. Transient Mode Case Studies412
4.2.1. Case 2413
414
The objective of the second case study is to evaluate the accuracy of our proposed adaptation method415
in transient conditions and evaluate how this may be affected by the initial map shape selected. Therefore,416
in this case study an additional model (Model II) is employed in which the compressor map sub-coefficients417
have been determined by fitting the data of another compressor map from PROOSIS, as described earlier418
23
in Section 3. The initial engine model where the compressor map generation was based on the map shown419
in Fig. 1 is going to be referred to as Model I for this case study.420
In contrast to the previous steady state case where the operating range was wide, in this transient case421
we are focusing on a much narrower range, i.e. from 100% to 90% of the nominal value of the rotational422
speed N . The reason for this selection is firstly due to the fact that the majority of industrial gas turbines423
used for power generation spend most of their lifetime within this operating range, and secondly because424
we want to compare different adaptation methods in a operating region where their accuracy seem to be of425
similar magnitude as shown in Fig. 13.426
It should be emphasized that the transient case sheds more light on the accuracy by which each adaptation427
method reconstructs a compressor map shape. Note that by default the acceleration and deceleration428
trajectories of the engine access a larger proportion of the map shape itself in comparison with the steady429
state operating points which are within a single running line of the map. The fuel flow rate for this case is430
scheduled accordingly and the fuel flow command is shown in Fig. 15. Simulation results for the specified fuel
0 20 40 60 80 1000.85
0.9
0.95
1
Time (s)
ff/ffdes Fuel Flow
Figure 15: Fuel flow schedule for the transient response.
431
flow schedule of the reference engine and the engine models after employing different adaptation methods432
are shown in Figs. 16 and 17.433
0 20 40 60 80 1000.94
0.96
0.98
1
Time (s)
T3/T
3des
Reference Engine LS MethodNLS Method Adapted Model IAdapted Model II
Figure 16: Normalized to design conditions compressor discharge temperature as predicted by various adaptation methods
during transient response.
As observed from Fig. 16, both Model I and Model II employing the proposed adaptation are in excellent434
agreement with the reference engine. It should be noted that although Model I and Model II are based on435
24
a different compressor map their prediction is identical. This is something expected for a couple of reasons436
that is going to be described below. The difference between the two scaling methods [3], [4] is not significant437
since the operating range is small and both have matched the targeted measurements with an appropriate438
level of accurately.439
The proposed adaptation is capable of following the transient response as seen from the acceleration and440
deceleration trajectories that are shown in Fig. 17 more accurately than the other scaling methods. A closer441
look at Fig. 17 reveals the effects that the different compressor map shapes employed in each Model I and442
Model II have on the accuracy of the prediction before the adaptation is employed. It becomes clear that443
Model II underperforms significantly in comparison with Model I before adaptation. This has to do solely444
with the difference in the initial compressor map shape selected, as the one of Model I is similar to the one445
used by the reference engine. This difference has been minimized by employing the adaptation procedure.446
0.9 0.92 0.94 0.96 0.98 1
0.85
0.9
0.95
1
mc/mcdes
πc/π
cdes
Reference Engine LS MethodNLS Method Unadapted Model IAdapted Model I Unadapted Model IIAdapted Model II Original Map
Figure 17: Transient trajectories on the compressor characteristic map for various adaptation methods.
The unconstrained nature of the optimization algorithm employed and the method by which the compres-447
sor map curves are fitted are the most important factors in matching any compressor map shape, independent448
of the initial map selected. The margin by which the 23 sub-coefficients of Model II have to be tuned to449
match the reference engine data is larger that the one of Model I, as seen from Fig. 18. Furthermore, the450
values of the sub-coefficients after adaptation are the same for both Models I & II. This confirms the fact451
that the [fminsearch] algorithm [21] was capable of converging to the same solution even if the initial values452
of the 23 sub-coefficients employed in Models I & II were different. It can be concluded that the proposed453
method, independent of the initial map shape, can successfully control the position, distribution, spacing454
and curvature of the elliptical curves passing through multiple data points. Hence compressor map shapes455
25
5 10 15 200.8
1
1.2
1.4
No of Sub-coefficients
Norm
alize
dV
alue
Initial Model I Tuning Model II Tuning
Figure 18: Initial and tuned map sub-coefficients for the adapted engine Models I & II.
can be effectively generated.456
In terms of the optimization process where a total of 100 discrete operating points were used for this case457
the process converged after 5000 iterations, 20000 function evaluations with the elapsed CPU time of 0.8 sec458
for the Model I. The difference in computation performance between Model I and Model II was negligible, as459
Model II performed 6400 iterations to reach the same tolerance criteria at an elapsed CPU time of 1.1 sec.460
On the other hand, the elapsed CPU time for the LS and NLS methods was 0.2 sec and 2.1 sec, respectively.461
The prediction error of each method for the transient case study is shown in Fig. 19. The prediction462
error in the compressor discharge pressure p3 and the temperature T3 for the proposed adaptation are in463
the range of -0.3% to 0.1%, as opposed to the LS and NLS methods that are both within the range of -1.2%464
to 1.1%.465
20 40 60 80 100
−1
0
1
Time (s)
Error
(%)
p3 LS p3 NLS p3 Proposed
T 3 LS T 3 NLS T 3 Proposed
Figure 19: Measurable parameters error of various adaptation methods during transient response.
For instance, the adapted Models I & II have an average error in T3 which is equivalent to 0.2K, whereas466
for the scaling method the average error is about 1K. A prediction error of 1% for scaling methods is467
acceptable in terms of engine performance prediction, but it should be once more noted that the operating468
range that the adaptation is performed is close to the design point of the engine where both methods perform469
accurately in principle. In the following subsection the degradations are injected in the reference engine so470
that the capabilities of each adaptation method are tested to their limits.471
26
4.2.2. Case 3472
473
The objective of this case study is to evaluate the capability of our proposed method to predict a474
component degradation. As described earlier in Section 2.3, the mass flow capacity and efficiency of the475
reference engine compressor are reduced by -5% and -2.5%, respectively. This percentage decrease represents476
a typical maximum rate of the compressor fouling for a gas turbine that can be partly recovered with off-line477
washing. Generally, given that steady state data of high quality are difficult to obtain, diagnosing the health478
of a gas turbine might be based on transient data. Although the former is computationally challenging it479
should be noted that transient behavior is much more sensitive to the degradation than the steady state.480
Consequently, transient data can provide a better insight when one is required to perform fault diagnosis481
and health monitoring during this operational mode.482
The fuel flow for the transient maneuver is identical to the one shown in Fig. 15. A total of 60 discrete483
operating points were used for the adaptation case. The engine model after adaptation matches the degraded484
reference engine measurable parameters at a different level of accuracy depending on the method employed.485
The engine model predictions are in good agreement with the reference engine for the degraded conditions486
as well. The deviations ∆Γc, ∆ηc used in Figs. 20 and 21 refer to the difference between the predicted and487
injected component degradation as defined in eq. (13), i.e. ∆Γc = ∆Γcpred −∆Γcinj . As can be observed,488
the errors are in the range of -0.2% to 0.2% for the proposed adaptation method. For the LS and NLS489
methods the errors are in the range of -6.0% to 2.3% and -2.2% to 1.0%, respectively.490
0 20 40 60 80 100−3
−2
−1
0
1
Time (s)
∆Γc
(%)
LS Method NLS Method Proposed Adaptation
Figure 20: Deviation in mass flow capacity as predicted by various adaptation methods during transient response.
0 20 40 60 80 100
−2
0
2
Time (s)
∆ηc
(%)
LS Method NLS Method Proposed Adaptation
Figure 21: Deviation in compressor isentropic efficiency as predicted by various adaptation methods during transient response.
27
Table 5: Adaptation results for various case studies (CS).
Performance Computation
Approach Mean Error (%) Mean Error (%) DI (%) Time (sec)
CS 1 CS 2 CS 3 CS 1 CS 2 CS 3
LS 3.0 0.9 67 0.1 0.2 0.18
NLS 1.2 0.8 88 0.4 2.1 1.6
Proposed 0.15 0.1 98 0.23 0.8 0.7
As expected, the error becomes more evident in the transient phase of the operation. Our employed491
adaptation method predicts the injected degradation with an average error of 0.08% for the mass flow492
capacity and 0.027% for the isentropic efficiency. On the other hand, the average error for the LS and the493
NLS methods is 0.6% and 0.2% for the mass flow capacity and 1.3% and 0.7% for the isentropic efficiency,494
respectively. The diagnostic index for our proposed method is 0.9894, which implies that the diagnosis is495
98.94% effective and in comparison with the LS and the NLS methods it is more accurate by 31% and 10%,496
respectively. The case study results demonstrate the promising prospect of our method when applied for a497
gas turbine performance diagnosis. The proposed adaptation process converged after 6000 iterations, 20000498
function evaluations with an elapsed CPU time of 0.9 sec.499
The maximum deviations of -5% in the compressor flow capacity and -2.5% in the isentropic efficiency are500
selected specifically in order to demonstrate the capability of our proposed approach to a more challenging501
diagnostic task than considering deviations of -1% and lower. However, it should be noted that the accuracy502
of our method remains unchanged even for smaller deviations of the component parameters. The results503
from all case studies (CS) are summarized as shown in Table 5.504
4.3. Techno Economic Case Study505
4.3.1. Case 4506
507
The primary objective of this case study is to assess the effects that each adaptation method have on508
the quality of engine performance information that are usually implemented by the gas turbine users for509
maintenance of a power plant. Generally gas turbine users implement various tools for estimating the health510
of gas turbine compressor in terms of the mass flow capacity and the efficiency. Depending on the severity511
of degradation that is predicted by the selected approaches, the plant has to be shut down and an off-line512
washing is carried out to recover the lost performance of the compressor and the engine.513
Although both on-line and off-line washing are performed on gas turbine compressors, the former is514
not accounted here. The reason is that the off-line washing has a higher impact on both recovering the515
28
compressor fouling and increasing the maintenance cost of the plant. The assumptions that are made for516
this case study are as follows:517
• The time frame for which a gas turbine operation is investigated is 12 months.518
• Only the compressor degradation due to the fouling is examined.519
• A typical constant rate of -1% drop in the compressor efficiency per month is assumed.520
• Off-line washing is performed once the estimated drop in the compressor efficiency is 1%.521
• After each off-line washing 95% of the lost compressor efficiency is recovered.522
• The gas turbine operates most of the time from 100% down to 90% of the nominal rotational speed523
N .524
• Within the above operating range a small proportion (20%) is considered as transient and the remaining525
(80%) as steady state.526
• A weighted average error is determined to account for the earlier assumption, i.e. εw = 0.8εs + 0.2εtr,527
where εs and εtr denote the average errors for the steady state and transient conditions, respectively.528
• The proposed adaptation method in conjunction with the scaling methods are employed.529
The number of off-line compressor washings that are suggested by each method is shown in Fig. 22 and530
is compared with the optimum number of washings that are required by the reference engine when adopting531
the maintenance strategy as encapsulated by the above assumptions.532
0 50 100 150 200 250 300 350
98
99
100
Time (days)
η c/η
cdes(%
)
Optimum Reference Engine LS Method
NLS Method Proposed Adaptation
Figure 22: Normalized to design conditions percentage of the compressor efficiency before and after off-line washings as suggested
by various adaptation methods. Local troughs represent the estimated drop of the compressor efficiency before performing
off-line washing. Peaks represent the recovered compressor efficiency after off-line washing.
It can be observed that the accuracy of the proposed adaptation method suggests 13 off-line washings533
for a one year period in comparison with the actual compressor degradation of the reference engine that can534
be optimally recovered with 12 washings only. The total number of washings suggested by the LS and the535
29
NLS methods is 26 and 21, respectively. Although the prediction accuracy of each method that is employed536
in this case study as shown in Fig 23 is in reasonable levels, their difference increases exponentially when537
it comes to the techno economics and the cost that each method implies in the maintenance of the power538
plant.539
0 50 100 150 200 250 300 350−1.5
−1
−0.5
0
Time (days)
∆η c
(%)
Optimum Reference Engine LS Method
NLS Method Proposed Adaptation
Figure 23: Percentage drop of compressor efficiency as estimated by various adaptation methods when adopting their suggested
number of off-line washings.
Let us make a reasonable assumption based on the work of Aretakis et al. [17] by considering the extra540
cost that each method implies in the maintenance of a gas turbine. Let us consider that each off-line washing541
costs $3000, which is reasonable for an aeroderivative gas turbine, and that the shutdown is about 3 hours.542
The cost corresponding to each method and its relative cost with respect to the optimum method of the543
reference engine are summarized in Table 6.544
Table 6: Cost of off-line compressor washings suggested by various adaptation method.
Approach Washings Cost ($)Cost rel. to
the Opt. (%)
Optimum 12 36,000 -
LS Method 28 78,000 216%
NLS Method 21 63,000 175%
Proposed
Adaptation13 39,000 108%
It is evident from Table 6 that the accuracy of our proposed method has the potential to reduce the545
maintenance cost associated only with off-line compressor washing, by 108% and 67% of the optimum cost546
in comparison with the existing LS and NLS methods, respectively. In practice, the cost associated with547
the compressor washing leads to several other associated costs such as shutdown, personnel, etc. that all548
contribute to the overall maintenance cost of a power plant. However, by using a group of assumptions the549
effects that the accuracy of various performance adaptation methods have in terms of plant economics are550
30
further amplified.551
As far as the practical aspects and limitations of our proposed method is concerned when used in real552
engines and real fault cases several considerations must be taken into account. Firstly, our proposed method553
can predict accurately the performance and health of a gas turbine for the range of operation that the initial554
model is adapted to. Therefore, one should initially adapt the engine model to a wide operating envelope of555
the engine. Feeding new operating points that belong to speed lines, not previously employed in the adap-556
tation, would only test the extrapolation capability of the map generation method and not the accuracy557
of the optimization for the diagnostic purposes. The proposed method does not consider variable geometry558
features such as the variable stator vanes (VSV), and how these affect the geometry of the compressor map559
generated. This is a limiting factor of our approach when one investigates a wide range of engine operations560
that utilize variable geometry compressor scheduling by the VSVs.561
Data corrections for measurement noise, humidity etc. should be accounted for before utilizing data562
smoothing. Data averaging for several engine performance measurements, such as the turbine and the power563
turbine exits, that rely on a set of instrumentation sensors might be employed. This facilitates the establish-564
ment of a good quality set of engine operating points to be utilized for the map generation and adaptation.565
The resulting adapted model forms the benchmark for further diagnostic analysis where deviations of the566
component parameters will be based upon this. However, one should consider the maintenance activity of567
an engine from the time of the first adaptation of the engine model up to the time that diagnosis is pur-568
sued. Meantime updating and refining the engine model should be performed. This process of continuously569
updating the engine model will improve the accuracy of the diagnosis significantly.570
The desirable features and performance capabilities of the proposed method motivate the inclusion of571
variable geometry characteristics to the compressor map generation and also the fitting and modelling of572
turbine maps; tasks that the authors are currently engaged in.573
5. Conclusions574
In this paper, a novel adaptation method is introduced that aims at improving the accuracy of the gas575
turbine performance prediction and diagnostics at both the steady state and transient operating conditions.576
The model coefficients that are obtained from the map generation procedure are optimized through a non-577
linear algorithm in order to match the targeted (healthy or degraded) measurements of a reference model578
with a compressor map that is available in the literature, working at off-design steady and transient con-579
ditions. The proposed method deals effectively with the nonlinear behavior of gas turbines away from the580
nominal operating points by tuning the compressor map shape resulting in accurate compressor degradation581
diagnostics.582
Application of the developed approach to a two shaft industrial gas turbine engine model demonstrates583
31
the following advantages and benefits. In comparison with earlier adaptation methods, our proposed strategy584
demonstrates the effectiveness that the compressor map has on the prediction accuracy of the engine model.585
The accuracy of the proposed method is independent of the similarity between the initial map shape selected586
and the targeted compressor map. The built-in nonlinear optimizer that is employed for this adaptation is587
effective in minimizing the prediction errors by adapting the compressor maps. The computational time for588
a typical multi-point steady state adaptation scenario is approximately 0.2 sec for 2000 function evaluations589
and 750 iterations with an average prediction error of 0.15%. Similarly, corresponding to the transient590
conditions and for 100 operating points the elapsed CPU time and the prediction error are 0.8 sec and591
0.1%, respectively. Our method is applied for predicting a compressor fouling degradation where the results592
demonstrate a diagnostic accuracy of 99.84% when 60 operating points of the reference engine in the transient593
mode are considered as the targeted measurements. In addition, the proposed method is capable of reducing594
the maintenance cost of a plant associated with the compressor washing ranging from 67% up to 108% in595
comparison with other existing adaptation methods.596
Our proposed adaptive performance method is a useful tool for progressively refining an engine model597
based on multiple sets of reference engine test data at both the steady state and the transient off-design598
operating conditions. The improved accuracy and efficient computational properties of our method have599
also demonstrated its potential capabilities for gas turbine diagnostics and reducing the maintenance cost.600
Therefore, implementation of our proposed method to any gas turbine performance simulation or as a601
condition monitoring and diagnostic tool could provide a more reliable and accurate information for gas602
turbine engines, and support the users in making informed decisions on managing efficiently their assets,603
increasing their availability and reducing their maintenance cost.604
Acknowledgements605
This publication was made possible by NPRP grant No. 4-195-2-065 from the Qatar National Research606
Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the607
authors. The authors would also like to acknowledge the constructive comments, and suggestions provided608
by the anonymous reviewers that greatly improved the quality of the article.609
References610
[1] I. Templalexis, P. Pilidis, V. Pachidis, P. Kotsiopoulos, Development of a 2-d compressor streamline curvature code, in:611
ASME Turbo Expo 2006: Power for Land, Sea, and Air, American Society of Mechanical Engineers, 2006, pp. 1005–1014.612
[2] Y. Yu, L. Chen, F. Sun, C. Wu, Neural-network based analysis and prediction of a compressor’s characteristic performance613
map, J. Appl. Energy 81 (1) (2007) 48–55.614
[3] C. Kong, S. Kho, J. Ki, Component map generation of a gas turbine using genetic algorithms, J. Eng. Gas Turbines Power615
128 (1) (2004) 92–96.616
32
[4] Y. G. Li, M. F. A. Ghafir, K. Huang, X. Feng, L. Wang, R. Singh, W. Zhang, Improved multiple point nonlinear genetic617
algorithm based performance adaptation using least square method, J. Eng. Gas Turbines Power 134 (3) (2012) 031701.618
[5] J. Kurzke, How to get component maps for aircraft gas turbine performance calculations, in: Proc. ASME Turbo Expo,619
1996.620
[6] G. Jones, P. Pilidis, B. Curnock, Extrapolation of compressor characteristics to the low-speed region for sub-idle perfor-621
mance modelling, in: Proc. ASME Turbo Expo, Vol. 2, Amsterdam, Netherlands, 2002, pp. 861–867.622
[7] V. Sethi, G. Doulgeris, P. Pilidis, A. Nind, M. Doussinault, P. Cobas, The map fitting tool methodology: gas turbine623
compressor off-design performance modeling, Journal of Turbomachinery 135 (6) (2013) 061010.624
[8] C. Drummond, C. Davison, Capturing the shape variance in gas turbine compressor maps, in: Proc. ASME Turbo Expo,625
Vol. 1, Orlando, USA, 2009.626
[9] K. Ghorbanian, M. Gholamrezaei, An artificial neural network approach to compressor performance prediction, J. Appl.627
Energy 86 (7) (2009) 1210–1221.628
[10] C. Kong, J. Ki, M. Kang, A new scaling method for component maps of gas turbine using system identification, J. Eng.629
Gas Turbines Power 125 (4) (2003) 979–985.630
[11] Y. G. Li, P. Pilidis, Ga-based design-point performance adaptation and its comparison with icm-based approach, J. Appl.631
Energy 87 (1) (2010) 340–348.632
[12] R. Ganguli, Gas Turbine Diagnostics: Signal Processing and Fault Isolation, CRC Press, 2012.633
[13] E. Tsoutsanis, Y. G. Li, P. Pilidis, M. Newby, Part-load performance of gas turbines: part 1 a novel compressor map634
generation approach suitable for adaptive simulation, in: Proc. ASME Gas Turbine India, Vol. 1, Mumbai, India, 2012,635
pp. 733–742.636
[14] PROOSIS, Propulsion Object-Oriented Simulation Software, see also http://www.proosis.com/ (2014).637
[15] E. Tsoutsanis, N. Meskin, M. Benammar, K. Khorasani, Dynamic performance simulation of an aeroderivative gas turbine638
using the matlab simulink environment, in: Proc. ASME IMECE, Vol. 4, San Diego, USA, 2013, p. V04AT04A050.639
[16] E. Tsoutsanis, Y. G. Li, P. Pilidis, M. Newby, Part-load performance of gas turbines: part 2 multi-point adaptation with640
compressor map generation and ga optimization, in: Proc. ASME Gas Turbine India, Vol. 1, Mumbai, India, 2012, pp.641
743–751.642
[17] N. Aretakis, G. Doumouras, I. Roumeliotis, K. Mathioudakis, Compressor washing economic analysis and optimization643
for power generation, J. Appl. Energy 95 (2012) 77–86.644
[18] J. Klapproth, M. Miller, D. Parker, Aerodynamic development and performance of the cf6-6/lm2500 compressor, in: Proc.645
4th International Symposium on Air Breathing Engines, Orlando, USA, 1979, pp. 243–249.646
[19] P. K. Zachos, I. Aslanidou, V. Pachidis, R. Singh, A sub-idle compressor characteristic generation method with enhanced647
physical background, J. Eng. Gas Turbines Power 133 (8) (2011) 081702.648
[20] A. J. Fawke, H. I. H. Saravanamuttoo, Digital computer methods for prediction of gas turbine dynamic response, Tech.649
rep., SAE Technical Paper (1971).650
[21] MATLAB, version 8.3 (R2014a), The MathWorks Inc., Natick, Massachusetts, 2014.651
[22] J. Lagarias, J. Reeds, M. Wright, P. Wright, Convergence properties of the nelder–mead simplex method in low dimensions,652
SIAM Journal on optimization 9 (1) (1998) 112–147.653
[23] Y. G. Li, P. Nilkitsaranont, Gas turbine performance prognostic for condition-based maintenance, J. Appl. Energy 86 (10)654
(2009) 2152–2161.655
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